Linearity of groups definable in o-minimal structures

Abstract

We consider an arbitrary o-minimal structure \(\mathcal {M}\) and a definably connected definable group G. The main theorem provides definable real closed fields \(R_1,\ldots ,R_k\) such that G / Z(G) is definably isomorphic to a direct product of definable subgroups of \(\mathrm{GL}_{n_1}(R_1),\ldots ,\mathrm{GL}_{n_k}(R_k)\), where Z(G) denotes the center of G. From this, we derive a Levi decomposition for G and show that [G, G]Z(G) / Z(G) is definable and definably isomorphic to a direct product of semialgebraic linear groups over \(R_1,\ldots ,R_k\).